A journey down the rabbit hole of Rubik's-style puzzles

For the most part, I still currently use the Beginner’s Method (as taught by RobH0629‘s very accessible and excellent tutorials) to solve a 3×3 cube. Although I’m amazed that I’ve gotten down to 1:09 using that method, I realize that I’ll need more advanced techniques to cut my times. For example, on even my fastest solves, the cross takes me an average of 15 seconds; with new techniques, I should be able to halve that.

When I say cross, I mean forming a cross/plus-sign in the bottom face (usually white) by placing the white/red, white/blue, white/orange, white/green edges with their white halves facing down and colored halves lined up with each side face’s center cube. Like so:

The Beginner’s Method starts by counter-intuitively placing the white edges in the TOP face — next to the yellow center, facing up. Next, each of those edges is put in the bottom face by (1) lining up the other color of each edge with the face of the same color (through U or U’ turns) and (2) turning the side face 180 degrees (R2, for example). Doing that for each of the four white edges FORCES proper relative positioning of the edges in the bottom face. The approach is brainless, but also highly inefficient.

Just the four 180° moves burn through more than half of the total moves required in virtually all cases. Indeed, as this study reveals, 99.95% of scrambles can be turned into a cross in 7 moves or less, with an average of 5.81 moves:

Moves

# Cases

Distribution

Cumulative

0

1

0.00%

< 0.01%

1

15

0.01%

0.01%

2

158

0.08%

0.09%

3

1,394

0.73%

0.82%

4

9,809

5.16%

5.99%

5

46,381

24.40%

30.39%

6

97,254

51.16%

81.55%

7

34,966

18.40%

99.95%

8

102

0.05%

100.00%

That sort of hyper-efficiency depends on look-ahead abilities far exceeding my own. But it does highlight just how inefficient the Beginner’s Method is. The trick, as I’ve learned from a few very good cross tutorials (and especially Badmephisto‘s), is to concentrate on relative positioning (to the exclusion of absolute positioning) of the four cross edges. That requires fundamentally understanding the color scheme of a standard cube’s side faces, which (with yellow up and white down) is:

clockwise: red, blue, orange, green

counter-clockwise: red, green, blue, orange

So long as the four cross edges are placed in the correct relative positions to one another, it takes only a turn of the D face to line up each edge with the proper face, as demonstrated below:

requires D

requires D

requires D’

requires D2

Although relative positioning requires far more brain power (at least for me!), it is far more efficient than placing seriatim the edges directly under the center color of the side faces. I’ve practiced maybe 50 cross solves with this technique, gaining a bit more fluency and fluidity with each. At an average of 20 seconds, these cross solves are still slower than with the Beginner’s Technique. But I can definitely see the opportunity for eventual time savings.